# Sparsity-certifying Graph Decompositions

## TL;DR

This paper introduces a new algorithm called the $(k,\, extell)$-pebble game with colors, providing a characterization of $(k, extell)$-sparse graphs and new methods for graph decomposition, with applications in rigidity theory.

## Contribution

The paper presents a novel colored pebble game algorithm that generalizes previous results and offers new sparse graph decompositions and proofs, enhancing understanding of graph sparsity.

## Key findings

- Provides a new characterization of $(k, extell)$-sparse graphs.
- Introduces a decomposition method certifying sparsity.
- Connects pebble game algorithms with existing sparse graph algorithms.

## Abstract

We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,\ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the $(k,\ell)$-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and Westermann and Hendrickson.